Andrew Heiss A. Jordan Nafa
Georgia State University University of North Texas
While the past two decades have been characterized by considerable progress in developing approaches to causal inference in situations where true experimental manipulation is either impractical or impossible, as is the case in much of political science, commonly employed approaches have developed largely within the frameworks of classical econometrics and frequentist non-parametrics (Blackwell and Glynn 2018). Unfortunately, these frameworks are limited in their ability to answer many of the questions scholars of international relations and comparative politics are often interested in since they rely heavily upon the assumption of long-run replication rather than quantifying uncertainty directly (Gill 1999; Gill and Heuberger 2020; Schrodt 2014; Western and Jackman 1994). In this article we develop a Bayesian approach to the estimation of marginal structural models for causal inference with cross-sectional time series and panel data. We assess the proposed models’ performance relative to existing procedures in a simulation study and two empirical examples, demonstrating that our approach performs well in terms of recovering the true parameter values while also lending itself to a more direct and intuitive interpretation. To ensure accessibility, we provide a flexible implementation of the proposed model in the R package brms (Bürkner 2017, 2018).

Bayesian Inference in Political Science

Introduced to political science in a series of articles by Bruce Western and Simon Jackman (Jackman 2000, 2004; Western 1998; Western and Jackman 1994), Bayesian inference provides several advantages over the frequentist paradigm, particularly for observational research in comparative politics and international relations where the logic of classical inference is often difficult to defend in practice (Gill 1999; Gill and Heuberger 2020; Schrodt 2014; Western and Jackman 1994). Paired with significant advances in computational power, the development of efficient Markov Chain Monte Carlo (MCMC) algorithms and user-friendly open-source software packages has made Bayesian inference widely accessible to researchers in the social sciences (Bürkner 2017, 2018; Goodrich et al. 2020). Despite its popularity in the development of measurement models (Claassen 2019, 2020; Clinton, Jackman, and Rivers 2004; Juhl 2018; Marquardt and Pemstein 2018), the application of Bayesian approaches to hypothesis testing remains relatively rare and is generally limited to circumstances where appropriate frequentist models cannot be applied such as in the presence of complete separation in discrete choice models (Gelman et al. 2008; Rainey 2016). Since some readers may be unfamiliar with the logic and principles of Bayesian methods, this section provides a brief overview of contemporary Bayesian inference and its advantages over the null hypothesis significance testing paradigm that remains dominant in political science.1

Following a series of contentious debates between Ronald Fisher, Jersey Neyman, Egon Pearson and other early twentieth century statisticians (Fisher 1930, 1935; Neyman and Pearson 1933a, 1933b), the null hypothesis significance testing (NHST) paradigm emerged from an attempt to merge two fundamentally incompatible approaches to scientific inference–Neyman and Pearson’s null hypothesis test and Fisher’s test of significance (Gill 1999). Neyman and Pearson’s null hypothesis test relies on the logic of modus tollens or proof by contradiction

Marginal Structural Models in a Bayesian Framework

\[ \begin{aligned} y_{i} & \sim \mathcal{N}(\mu, \sigma)^{\tilde{w_{i}}}\\ \mu & = \alpha + X_{n}\beta_{k} + \sigma\\ \textit{where}\\ \alpha & \sim \mathcal{Student \, T}(\nu_{\alpha}, \, \mu_{y}, \, \sigma_{y})\\ \beta_{k} & \sim \mathcal{MVN}(0, \, \Sigma_{\beta})\\ \sigma & \sim \mathcal{Student \, T}_{+}(3, \, 0, \, \sigma_{y})\\ \end{aligned} \]

References

Blackwell, Matthew, and Adam N. Glynn. 2018. “How to Make Causal Inferences with Time-Series Cross-Sectional Data Under Selection on Observables.” American Political Science Review 112: 1067–82.
Bürkner, Paul-Christian. 2017. “Brms: An r Package for Bayesian Multilevel Models Using Stan.” Journal of Statistical Software 80: 1–28.
———. 2018. “Advanced Bayesian Multilevel Modeling with the r Package Brms.” The R Journal 10: 395–411.
Claassen, Christopher. 2019. “Estimating Smooth Country Year Panels of Public Opinion.” Political Analysis 27: 1–20.
———. 2020. “Does Public Support Help Democracy Survive?” American Journal of Political Science 64: 118–34.
Clinton, Joshua, Simon Jackman, and Douglas Rivers. 2004. “The Statistical Analysis of Roll Call Data.” American Political Science Review 98: 355–70.
Fisher, R. A. 1930. “Inverse Probability.” Mathematical Proceedings of the Cambridge Philosophical Society 26: 528--535.
———. 1935. “The Logic of Inductive Inference.” Journal of the Royal Statistical Society 98(1): 39–82.
Gelman, Andrew, Aleks Jakulin, Maria Grazia Pittau, and Yu-Sung Su. 2008. “A Weakly Informative Prior Distribution for Logistic and Other Regression Models.” Annals of Applied Statistics 2: 1360–83.
Gelman, Andrew, and Cosma Rohilla Shalizi. 2012. “Philosophy and the Practice of Bayesian Statistics.” British Journal of Mathematical and Statistical Psychology 66(1): 8–38.
Gill, Jeff. 1999. “The Insignificance of Null Hypothesis Significance Testing.” Political Research Quarterly 52: 647–74.
Gill, Jeff, and Simon Heuberger. 2020. “Bayesian Modeling and Inference: A Post-Modern Perspective.” In The SAGE Handbook of Research Methods in Political Science and International Relations, eds. Luigi Curini and Robert Franzese. London, UK: SAGE, 961–84.
Goodrich, Ben, Jonah Gabry, Imad Ali, and Sam Brilleman. 2020. “Rstanarm: Bayesian Applied Regression Modeling via Stan.” https://mc-stan.org/rstanarm.
Jackman, Simon. 2000. “Estimation and Inference via Bayesian Simulation: An Introduction to Markov Chain Monte Carlo.” American Journal of Political Science 44: 375–404.
———. 2004. “Bayesian Analysis for Political Research.” Annual Review of Political Science 7: 483–505.
Juhl, Sebastian. 2018. “Measurement Uncertainty in Spatial Models: A Bayesian Dynamic Measurement Model.” Political Analysis 27(3): 302–19.
Marquardt, Kyle L., and Daniel Pemstein. 2018. “IRT Models for Expert-Coded Panel Data.” Political Analysis 26(4): 431–56.
McElreath, Richard. 2020. Statistical Rethinking: A Bayesian Course with Examples in r and STAN. 2nd ed. Chapman; Hall/CRC.
Neyman, Jerzy, and Egon S. Pearson. 1933a. “On the Problem of the Most Efficient Test of Statistical Hypotheses.” Philosophical Transactions of the Royal Statistical Society 231(694-706): 289–337.
———. 1933b. “The Testing of Statistical Hypotheses in Relation to Probabilities a Priori.” Mathematical Proceedings of the Cambridge Philosophical Society 29: 492–510.
Rainey, Carlisle. 2016. “Dealing with Separation in Logistic Regression Models.” Political Analysis 24(3): 339–55.
Schrodt, Philip A. 2014. “Seven Deadly Sins of Contemporary Quantitative Political Analysis.” Journal of Peace Research 51: 287–300.
Western, Bruce. 1998. “Causal Heterogeneity in Comparative Research: A Bayesian Hierarchical Modelling Approach.” American Journal of Political Science 42(4): 1233.
Western, Bruce, and Simon Jackman. 1994. “Bayesian Inference for Comparative Research.” American Political Science Review 88: 412–23.

  1. It is worth noting here that there is relatively little agreement among statisticians regarding the application of Bayesian estimation and the meaning of the term often varies substantially within and across disciplines. Our definition herein follows that of Gelman and Shalizi (2012) and McElreath (2020) that views priors as assumptions about the universe of plausible effect sizes and emphasizes aggressive model checking as opposed to the philosophical Bayesian view of priors as entirely subjective beliefs.↩︎